Question

Use the given graph to determine the limit, if it exists. A coordinate graph is shown with a horizontal line crossing the y axis at six that ends at the open point 2, 6, a closed point at 2, 1, and another horizontal line starting at the open point 2, negative 3. Find limit as x approaches two from the left of f of x. and limit as x approaches two from the right of f of x..

Answer 1

The limit of the function does not exists.

Explanation:

From the graph it is noticed that the value of the function is 6 from all values of x which are less than 2. At x=2, the line y=6 has open circle. It means x=2 is not included.

For x<2

`f(x)=6`

The value of the function is -3 from all values of x which are greater than 2. At x=2, the line y=-3 has open circle. It means x=2 is not included.

For x>2

`f(x)=-3`

The value of y is 1 at x=2, because of he close circles on (2,1).

For x=2

`f(x)=1`

Therefore the graph represents a piecewise function, which is defined as

`f(x)=\begin{cases}6& \text{ if } x<2\\ 1& \text{ if } x=2 \\ -3& \text{ if } x>2 \end{cases}`

The limit of a function exist at a point a if the left hand limit and right hand limit are equal.

`lim_(x\rightarrow a^-)f(x)=lim_(x\rightarrow a^+)f(x)`

The function is broken at x=2, therefore we have to find the left and right hand limit at x=2.

`lim_(x\rightarrow 2^-)f(x)=6`

`lim_(x\rightarrow 2^+)f(x)=-3`

`6\\eq-3`

Since the left hand limit and right hand limit are not equal therefore the limit of the function does not exists.